I was tattooed in the midst of my OW certification trip about 10 years ago, so here's my experience. First, as has been mentioned, exposing your tattoo to the sun (ever) will likely cause it to fade. The extent depends on a lot of things, including location, skin type and texture, etc. It's all very individual. I discussed the issue of tattooing in general with a tattoo artist who was a former medical professional, a doctor of dermatology. Basically, the general consensus is that if you go out in the sun, it's going to fade, regardless how soon the sun exposure is after you got the tattoo. If you're a sun lover like me, it's one of the known risks of getting a tattoo in the first place, and is expected. However, getting a sun burn on a fresh wound (like a tattoo) WILL inhibit healing.
Now, sun issues aside, the biggest risk of infection is NOT the sea water itself (the following doesn't apply to deep or puncture wounds... This is specifically related to abrasions and tattoo type wounds). Frequently irrigating any wound with fresh, moving water isn't likely to cause an infection. This would apply to swimming and diving, regardless the length of the (reasonable) exposure. Now, here's a bit of info on how to keep your tattoo looking good even though you want to get wet.
1. Let it heal through the first scabbing stage. The process of the first scabbing stage is "setting" the ink. Significant moisture will loosen the scabs and cause them to pull free. This will take some of the ink with, causing a "blotchy" appearance. You are also at a higher risk of infection since the scab is a protective covering of the wound. Once removed prematurely, it's a fresh wound all over again.
2. Your artist likely told you not to moisturize your tattoo with Vaseline. Don't. However, DO apply a layer of Vaseline to a fresh tattoo before you get in the water. This is a protective coating, like what the artist may have applied before he sent you out the door. It keeps the water out. There are other products that will work (moisture barrier products designed for burn victims and used in the medical industry) but this is the most common product that's appropriate. You will wash it off with warm soapy water as soon as you are done with your dive. Don't use anything medicated for this, ie no neosporin or the like.
3. Cover your tattoo with a CLEAN piece of clothing NOT COTTON. A polyester or nylon or similar type of shirt would be appropriate. Something that is close fitting. This is to reduce RUBBING, and has nothing to do with sun exposure. You are trying to prevent any type of accidental bumping or friction on the tattoo. Also to keep the Vaseline from coming off too quickly. If you are wearing a wet suit, having a shirt on is even more important. Some of the ink and any residual skin or scabbing will almost positively come off and transfer to the shirt. Please don't wear a rented or borrowed wet suit over a tattoo without covering it. The residual skin and residue is just nasty if it's not your gear.
4. Wash the tattoo with warm soapy water as soon as you get out of the water. No going shopping first, no hanging out at the beach first, no nothing first. Just go wash it. Then leave it uncovered. It needs to dry out.
This was what I did 10 years ago, and with every tattoo I've gotten since. Sometimes my job requires me to be in the water, but I do like tattoos

it's worked out quite well. The only time I had an issue with any of my tattoos was scabbing pulling out after wearing combat boots for hours on end during the summer. Most of my tattoo wound up affixed to my sock. Had I used the above process, it would have worked out better on that one. (The tattoo was on my ankle). Had it touched up and it's great now.
After all that, infection is always a possibility. Stay healthy, and if you see something oozing or looking funky, go see your artist or a doctor. You said that you heal quickly though, so you're probably not prone to infection. I'm the same way.
---------- Post added October 10th, 2013 at 11:21 AM ----------
Also, good luck on your ink. I'm planning my "big one" but a big dive trip is going to come first!